Dynamics Of Some Hgh-order Rational Difference Equations

With the development of science and technology, the difference equations are widely used in biological, economic and physical fields and so on. It is a clear fact that the non-linear difference equations exhibit more complicated dynamic behavious with respect to the linear case. Rational difference equations, as a special class of nonlinear differ-ence equations, whose study has gradually become an important branch of difference dynamic system, and it has a very high theoretical and applicable values.The main content of this dissertation as follows:In chapter 1, we introduce the developments and research significance of difference equations, and give the basic definitions and preliminaries of this paper.In chapter 2, we discuss a kind of difference equation with order k+2. By using the stability theory and convergence theorem, we obtained the local stability and global asymptotic stability of the positive equilibrium. Also, we analyze the periodicity and boundedness character of the solutions, with the method of semicycle analysis investi-gated the semicycle character of the solutions and invariant interval of the equations.In chapter 3, we discuss a kind of rational difference equations with order three. The formulas of solutions with nonzero initial values are obtained, and the proofs are completed. Further, based on the formulas of the solutions, the necessary and sufficient conditions are obtained, which guarantee the existence of periodic solutions and anti-periodic solutions.In chapter 4, we discuss a kind of rational difference equations with order four. The formulas of solutions with nonzero initial values are obtained and the periodic nature are analyzed. At last, the limit form of solutions are investigated.